Problem 34 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-2,-5) and (6,-5)

Step-by-Step Solution

Answer

The equation of the line in point slope form is y = -5 and in slope intercept form is also y = -5.

Step 1: Identify the Coordinates of the Points

The exercise gives two points through which the line passes: Point 1 as (-2, -5) and Point 2 as (6, -5).

Step 2: Calculate the Slope

The slope is calculated as the change in y divided by the change in x. Here both points have the same y-coordinate(-5), so the change in y is 0, making the slope 0.

Step 3: Write the Point-slope form

With a slope of 0 and using Point 1 coordinates for (x1, y1), the equation in point-slope form becomes (y - (-5)) = 0*(x - (-2)), which simplifies to y = -5.

Step 4: Write the Slope-intercept form

The slope is 0 and y-intercept is the y-coordinate of either point, so the equation in slope-intercept form is y = 0*x - 5, which simplifies to y = -5.

Key Concepts

Point-Slope Form

Linear equations can be represented in several forms, and the point-slope form is one of them. It is especially useful when you know a point on the line and the slope.
The general formula for point-slope form is given as: \[ y - y_1 = m(x - x_1) \] Here, \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. Using the point-slope form, you can directly write the equation of the line provided you have these two pieces of information.
In our exercise, we identified the point \((-2, -5)\) and calculated the slope to be 0. Using these, we plug into the formula:

• \( y - (-5) = 0(x - (-2)) \) simplifies to \( y + 5 = 0 \)
• Further simplifies to \( y = -5 \)

This indicates that our line is horizontal, as expected with a slope of zero.

Slope-Intercept Form

Another convenient way to express a linear equation is in slope-intercept form. This format emphasizes the slope and the point where the line crosses the y-axis, called the y-intercept.
The slope-intercept form is written as: \[ y = mx + b \] Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept.
For our problem, since the slope is 0 and since the line passes through the y-value of -5 (horizontal line), the equation becomes:

• \( y = 0 \times x - 5 \)
• Simplifies to \( y = -5 \)

This equation confirms that wherever you pick the value of \( x \), \( y \) remains \(-5\), signifying a horizontal line.

Slope Calculation

Understanding how to calculate the slope is a crucial step in handling linear equations. The slope is a measure of how "steep" the line is and is commonly represented by \( m \).
It involves comparing the change in y-values to the change in x-values between two points on a line. The formula to calculate slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula provides a ratio to describe the tilt of the line.
In our specific example, both given points, \((-2, -5)\) and \((6, -5)\), share the same y-value. Hence, \( y_2 - y_1 = 0 \), leading to a slope \( m = 0 \).

• When the slope is 0, the line is horizontal.
• No matter how \( x \) changes, \( y \) remains constant.

Slope 0 means the line is parallel to the x-axis, a concept frequently encountered in algebra.